AOE network and Key Path (1)

1. The AOE network is closely related to the AOV network. If a directed edge is used in a DAG to represent various activities of a project, the directed edge is used to indicate the activity duration, and the vertex is used to represent the event, this directed graph is called an Activity On Edge (AOE network;

2. Because there is only one start point and one completion point for the entire project, the point of the start point (inbound degree is 0) is the source point, and the end point (outbound degree is 0) is the sink point;

3. The AOE network is very useful in some aspects (such as engineering estimation). For example, the AOE network can understand the following two points:

(1) How long does it take to complete the entire project (assuming there is no ring in the Network )?

(2) what activities should be accelerated to shorten the time required to complete the project?

4. In the AOE network, some activities can be performed in parallel. There may be more than one directed path from the source point to each vertex and from the source point to the sink point. These paths may have different lengths. Although the total time required for completing each activity on different paths is different, the entire project is completed only when all the activities on each path are completed. Therefore, the time required to complete the entire project depends on the longest path length from the source point to the sink point, and the time required is the sum of the duration of all activities on the path. The path with the longest length becomes a key path;

5. To find the key path, you must find the key activities (the activities completed by the entire project will be affected if they are not completed on schedule). The following defines several quantities related to computing key activities:

(1) the earliest possible start time of event Ei, recorded as Ee [I], is the longest path length from the source point E0 to the vertex Ei;

(2) The latest allowable start time of event Ei, which is recorded as El [I], is to ensure that the en-1 of the event is completed at the Ee [n-1] Time, the Start Time (El [I] = Ee [n-1]-(the longest path length from Ei to En-1 );

(3) the earliest possible start time of the activity ak, which is recorded as e [k]. If the accesskey is set to an edge <Ei, Ej>, e [k] = Ee [I];

(4) The latest allowed start time of an activity ak, which is recorded as l [k]. It is the latest allowed start time without causing any delay, that is, l [k] = El [j]-dur (<Ei, Ej>), wherein dur (<Ei, Ej>) is the time required to complete the active ak, indicates the weight of an edge <Ei, j>;

6. l [k]-e [k] indicates the earliest possible start time and the latest allowable start time margin of the active ak, which is also called the relaxation time, l [k] = e [k] indicates that the active ak has no time margin and is a key activity;

7. The purpose of analyzing the key paths is to estimate each activity from the source E0 and identify the key activities that affect the progress of the entire project so as to arrange the work scientifically;

8. To identify key activities, we need to obtain the e [k] and l [k] of each activity to determine whether they are equal; to obtain e [k] and l [k], we need to first obtain the Ee [I] and El [I] from the source E0 to each vertex Ei. The Recursive Formulas for Ee [I], El [I], e [k], and l [k] are described as follows:

(1) Ee [I]: Start from Ee [0] = 0, forward recurrence:

(2) El [I]: from El [n-1] = Ee [n-1], reverse recursion:

(3) e [k] and l [k]: Set the weighted directed edge of the active accesskey to <Ei, Ej>, and the duration of the accesskey is dur (<Ei, if yes, e [k] = Ee [I]; l [k] = El [j]-dur (<Ei, Ej> );

9. According to the above analysis, the algorithm for calculating the Key Path is as follows:

(1) input the directed edge of the m strip weight to establish the structure of the adjacent table;

(2) starting from E0, Ee [0] = 0. Ee [I], I =,..., n-1 of each vertex are calculated in the order of topological order. If the number of loops in the topological sorting is less than the number of vertices n, it indicates that a directed ring exists in the network and the Key Path cannot be obtained;

(3) starting from the sink point En-1, let El [n-1] = Ee [n-1], calculate the El [I], I = n-2 of each vertex in the order of inverse topological order, n-3 ,..., 0;

(4) Calculate e [k] and l [k] of each directed edge based on Ee [I] and El [I] of each vertex;

(5) For each edge in the network, if e [k] = l [k] is satisfied, it is a key activity; find and output all key activities;

Ps: I 've written so much related knowledge... I'm exhausted. Next, the algorithm for specific implementation of key paths ~